Question

Saturated ammonia vapor at \(25^{\circ} \mathrm{C}\) condenses on the outside of a 2 -m-long, \(3.2\)-cm-outer-diameter vertical tube maintained at $15^{\circ} \mathrm{C}\(. Determine \)(a)\( the average heat transfer coefficient, \)(b)$ the rate of heat transfer, and \((c)\) the rate of condensation of ammonia. Assume turbulent flow and that the tube diameter is large, relative to the thickness of the liquid film at the bottom of the tube. Are these good assumptions?

Short answer

Answer: The average heat transfer coefficient is approximately 0.0470 W/m²K, the rate of heat transfer is approximately 0.0942 W, and the rate of condensation of ammonia vapor is approximately 6.87 x 10⁻⁸ kg/s. The assumption of turbulent flow is valid, as the Grashof number is approximately 2.17 × 10^9. The assumption of large tube diameter compared to liquid film thickness is considered reasonable in most condensation applications, although there is not enough evidence in this specific case to confirm or refute this assumption.

Step-by-step solution

Given data

Given data are: - Length of tube (L) = 2 m - Tube's outer diameter (D) = 3.2 cm = 0.032 m - Liquid film thickness (δ) is assumed to be small relative to tube diameter - Inlet temperature of ammonia vapor (T_vapor) = 25°C - Tube's surface temperature (T_wall) = 15°C - Assume turbulent flow

Calculate the temperature difference

ΔT = T_vapor - T_wall = 25°C - 15°C = 10°C

Calculate the Nusselt number using Nusselt relation

The Nusselt number (Nu) is used to determine the average heat transfer coefficient. For condensation, we can use the Nusselt relation for a vertical tube given by: Nu = 0.725 * (Pr * Gr)^(1/4) where Pr is the Prandtl number and Gr is the Grashof number. First, we need to find the Prandtl and Grashof numbers.

Find the properties of ammonia

To find these values, refer to a thermodynamic properties table for ammonia at T = 25°C (298 K): - Density of the liquid ammonia (ρ_l) = 602.3 kg/m³ - Density of ammonia vapor (ρ_v) = 1.63 kg/m³ - Thermal conductivity of ammonia (k) = 0.00187 W/mK - Kinematic viscosity of the vapor (ν) = 15.84 x 10⁻⁶ m²/s - Heat of vaporization (h_fg) = 1370 kJ/kg

Calculate the Prandtl number (Pr)

Pr is the ratio of momentum diffusivity and thermal diffusivity. Pr = ν / α, where α is the thermal diffusivity. By referring to the ammonia property table, we can find α = 17.63 x 10⁻⁶ m²/s. So Pr = 15.84 x 10⁻⁶ / 17.63 x 10⁻⁶ = 0.898

Calculate the Grashof number (Gr)

Gr is a measure of the ratio of buoyancy to viscous force, and is given by: Gr = (g * L^3 * (ρ_l - ρ_v) * ΔT) / (T_vapor * ν^2) Using the given values and the properties obtained in previous steps, we can calculate Gr: Gr = (9.81 * 2^3 * (602.3 - 1.63) * 10) / (298 * (15.84 x 10⁻⁶)^2) ≈ 2.17 × 10^9 Now, we can calculate the Nusselt number using the Nusselt relation. Nu = 0.725 * (0.898 * 2.17 x 10^9)^(1/4) = 50.42

Calculate the average heat transfer coefficient (h)

The average heat transfer coefficient is given by: h = (Nu * k) / L Substituting the values: h = (50.42 * 0.00187) / 2 ≈ 0.0470 W/m²K So, the average heat transfer coefficient is approximately 0.0470 W/m²K.

Calculate the rate of heat transfer (Q)

The rate of heat transfer is given by: Q = h * π * D * L * ΔT Substituting the values: Q = 0.0470 * π * 0.032 * 2 * 10 ≈ 0.0942 W So, the rate of heat transfer is approximately 0.0942 W.

Calculate the rate of condensation of ammonia (m_dot)

The rate of condensation (mass flow rate) is given by the following expression: m_dot = Q / h_fg Substituting the values: m_dot = 0.0942 / (1370 * 10^3) ≈ 6.87 x 10⁻⁸ kg/s So, the rate of condensation of ammonia is approximately 6.87 x 10⁻⁸ kg/s.

Discussing the assumptions

Are these good assumptions? - Turbulent flow assumption: With a Grashof number of approximately 2.17 × 10^9, the flow can be considered turbulent, so this assumption is valid. - Large tube diameter compared to liquid film thickness: Based on the given data and calculations, we don't have enough evidence to indicate whether this assumption is valid or not. However, in most condensation applications, the film thickness is thin enough so that this assumption can be considered reasonable. Finally, the average heat transfer coefficient is approximately 0.0470 W/m²K, the rate of heat transfer is approximately 0.0942 W, and the rate of condensation of ammonia vapor is approximately 6.87 x 10⁻⁸ kg/s.